Characterization of single nanoparticles

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by Steven Jones

BSc, University of Victoria, 2015 A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of MASTER OF APPLIED SCIENCE

in the Department of Electrical and Computer Engineering

 Steven Jones, 2016 University of Victoria

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Supervisory Committee

Characterization of Single Nanoparticles by

Steven Jones

BSc, University of Victoria, 2015

Supervisory Committee

Dr. Reuven Gordon, (Department of Electrical and Computer Engineering) Supervisor

Dr. Geoffrey Steeves, (Department of Physics and Astronomy) Outside Member

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Abstract

Supervisory Committee

Dr. Reuven Gordon, (Department of Electrical and Computer Engineering)

Supervisor

Dr. Geoffrey Steeves, (Department of Physics and Astronomy)

Outside Member

Optical trapping is a method which uses focused laser light to manipulate small objects. This optical manipulation can be scaled below the diffraction limit by using interactions between light and apertures in a metal film to localize electric fields. This method can trap objects as small as several nanometers. The ability to determine the properties of a trapped nanoparticle is among the most pressing issues to the utilization of this method to a broader range of research and industrial applications. Presented here are two methods which demonstrate the ability to determine the properties of a trapped nanoparticle.

The first method incorporates Raman spectroscopy into a trapping setup to obtain single particle identification. Raman spectroscopy provides a way to uniquely identify an object based on the light it scatters. Because Raman scattering is an intrinsically weak process, it has been difficult to obtain single particle sensitivity. Using localized electric fields at the trapping aperture, the Raman integrated trapping setup greatly enhances the optical interaction with the trapped particle enabling the required sensitivity. In this work, the trapping and identification of 20 nm titania and polystyrene nanoparticles is demonstrated.

The second method uses an aperture assisted optical trap to detect the response of a magnetite nanoparticle to a varying applied magnetic field. This information is then used to determine the magnetic susceptibility, remanence, refractive index, and size distribution of the trapped particle.

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List of Figures

Figure 1 – Kretschmann configuration of a surface plasmon resonance sensor. Left: schematic diagram of the measurement apparatus. Monochromatic light is focused onto a thin gold film via a prism. Surface plasmons are excited for a specific angle of incidence corresponding to an absorption peak as measured by the detector. As the analyte bonds to the functionalized surface of the gold, the effective index of the dielectric changes causing a shift in the plasmon resonance. Right: the angle of incidence corresponding to surface plasmon excitation as a function of time as the analyte if injected into the channel, saturates the sensing region, and is subsequently removed. ... 4 Figure 2 – Diagram illustrating the preparation and operation of a localized surface plasmon resonance (LSPR) based biosensor based on changes in the surrounding dielectric permittivity. a) and b) metal nanoparticles are deposited onto a substrate. c) nanoparticles are functionalized to bind with desired analyte. d) analyte binds to nanoparticles causing a frequency shift in the plasmonic resonance – shown in e). This figure has been reprinted with permission from22_{. ... 7}

Figure 3 – Dipole moment induced in a subwavelength dielectric sphere by an applied electric field. Notice the electric field lines are parallel within the sphere, i.e. constant E-field. ... 14 Figure 4 – Radiation pattern of dielectric sphere (dipole). The radial distance from any point on the surface of the wavefront (red surface) to the origin, is proportional to the intensity scattered in that direction. ... 15 Figure 5 – Lycurgus cup, a Roman artifact dating back to the 4th_{century AD. The glass}

contains trace amount of silver and gold nanoparticles which exhibit plasmon resonances, thereby affecting the transmission and reflection of light. The image on the left shows the cup transmitting light, while the figure on the right is viewed mainly through reflection. Reprinted with permission from71. ... 19 Figure 6 – Dispersion relation for metal-dielectric interface using the Drude model7 (𝜖∞ = 1, 𝜖𝑟, 𝑧 > 0 = 1). Left figure indicates the ideal lossless case, while the figure on the right includes losses (𝛾 = 0.1). The solid blue lines correspond to the real part of the propagation wave vector while the dashed blue lines are the imaginary part. The red line indicates the lossless surface plasmon polaritons frequency, and the solid black line is the light line showing the dispersion relation for a plane wave propagating in a vacuum (𝜖𝑟 = 1). .... 24 Figure 7 – Coupling between nearby nanoparticles and relative field enhancement with respect to electric field orientation. a) shows two small metallic particles with their relative position vector perpendicular to the applied electric field and negligible coupling. b) indicates the case where the nanoparticles are orientated in parallel with the applied electric field and a large electric field enhancement is observed between the two particles, known as a plasmonic “hot spot”. ... 26

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Figure 8 – Plasmonic resonances for gold nanoparticles as function of interparticle distance. Left: average optical cross section for coupled gold nanoparticles (20 nm spheres – top/red, ellipsoids with aspect ration = 2 – bottom/blue). Right: total cross section normalized to physical cross section for gold nanospheres and ellipsoids in various dielectric mediums. Reprinted with permission from22. ... 26 Figure 9 – a) graph showing the forces acting on a particle in an optical trapping configuration for a Gaussian beam profile. The point x = 0 at the intersection of the axis, coincides with the centre of the focused beam waist (dashed line in c)). b) a simplified force vector diagram of the ray-model approximation to trapping forces, the length of each vector indicates the relative intensity of that interaction. c) a schematic diagram of an optical trapping configuration. The circles are particles near the beam waist showing relative intensities of scattering, gradient, and net forces for each particle. The particle just above the dashed line indicates the stable trapping position as indicated in a). ... 29 Figure 10 – Illustrative example of transmission change through a subwavelength aperture upon dielectric loading which increases the effective refractive index. a) and b) show the transmission through the aperture without and with an embedded particle respectively. c) indicates the transmission curves as a function of wavelength. ... 32 Figure 11 – Transmission properties of cylindrical holes milled in suspended gold films. a) scanning electron microscope image of subwavelength aperture in gold film. b) transmission spectra at normal incidence for hole diameter d = 270 nm and for hole depth h as indicated. Reprinted with permission from95. ... 33 Figure 12 – Classical approximation to a diatomic molecule. ... 35 Figure 13 – Energy level diagram (left) for a diatomic molecule indicating electronic, vibrational, and rotations energy states. The distribution in energy of these vibrational states follows a Morse potential as indicated on the right. ... 38 Figure 14 – Normal modes of vibration for a CO2 molecule. Note that vibrational modes 3

and 4 correspond to the same molecular vibration rotated by 90 ∘20_{. ... 41}

Figure 15 – Polarizability changes of CO2 molecule as it goes through its normal modes of

vibrations20. Ellipsoids indicate polarizability of molecule for the configurations shown below. Numbers correspond to the vibrational modes shown in Figure 14. Only vibrational mode 1 in Raman active. ... 42 Figure 16 – Energy level diagram (left) of a molecule undergoing Raman scattering. The blue 𝑣’s indicate vibrational energy states and the black n’s are for electronic energy states. Dashed lines indicate a virtual energy state. On the right is an illustration of a hypothetical Raman spectra indicating the shift in frequency (energy) of inelastic scattering events corresponding to Stokes and anti-Stokes spectral peaks. The central peak labeled “Rayleigh” corresponds to elastic scattering events and has the same frequency as the excitation source. ... 44

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Figure 17 - The basic design of the optical trapping microscope used for the work in this thesis. ... 47 Figure 18 – Scanning electron microscope image of a dual nanohole aperture used in trapping. ... 49 Figure 19 – Simulation of electric field strength for a dual nanohole aperture. a) and b) show the z-profile of the aperture along the cusps (𝑦 = 0), while c) shows the xy-profile. Two different gradients for the z-profile are considered, a) for the case of a straight profile, and b) for a sloped cusp. Notice the large electric field enhancement at the cusps of the trapping aperture indicating the energetically favourable trapping location. ... 49 Figure 20 – Overview of the procedure used to prepare a microscope slide for nanoparticle trapping. An adhesive microwell spacer is attached to a clean #0 glass coverslide. Next, the nanoparticle solution is inserted into the microwell, and finally the gold coated coverslide containing the trapping aperture is placed on top, creating a sealed microwell containing the nanoparticles. ... 51 Figure 21 – Screenshot showing the camera image of the sample during alignment. Figure a) shows the circular fiduciary marker that is used to locate the trapping aperture, the DNH trapping aperture is the bright spot in the centre of the ring. Figure b) shows the trapping laser aligned with the DNH aperture. ... 53 Figure 22 – Schematic diagram of trapping setup used for integrated single particle Raman spectroscopy. The main modifications to the original setup (Figure 17) are labeled. This includes a 671 nm laser, 685 nm dichroic (long pass), a 671nm notch filter, and a spectrometer. ... 56 Figure 23 – Characteristic trapping event of 20 nm polystyrene recorded with the Raman integrated trapping setup. Blue line indicates raw transmission data while the green line is the data with a Savitzky-Golay filter applied. ... 57 Figure 24 – Comparison between single particle Raman spectra for single particle (red) and bulk (blue) 20 nm TiO2 nanoparticles. a) shows the raw reference-subtracted Raman

spectra observed during trapping with an acquisition time of 1 minute, averaged over 18 spectra, with a Savitzky-Golay filter applied. b) is the same spectra as a) but numerically adjusted to account for attenuation in the original spectra due to the dichroic filter and to adjust the slope of the single particle spectra to match that of the bulk solution. Insets show the full spectrum from 50 to 1700 cm-1. ... 59 Figure 25 – Single particle (red) and bulk (blue) Raman spectra for 20 nm polystyrene nanoparticles. The single particle spectrum is based on an average of 5 spectra with an acquisition time of 5 minutes each, and a Savitzky-Golay filter applied. ... 61

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Figure 26 – Raman difference-spectra for multiple trapping events of 20 nm TiO2

nanoparticles. The black lines indicate the spectrum observed with no trapped particle. The blue lines are for the case of one trapped particle, and the green lines are for two trapped particles. Thin lines indicate raw spectra while bold lines are averaged and filtered. ... 62 Figure 27 – Optical trapping with magnetic field setup (OTMF). The electromagnet used to probe the nanoparticles was placed near the trapping site but external from the trapping setup. The orientation is such that the applied magnetic force works to pull trapped magnetic nanoparticle out of the trap, towards the microwell. ... 67 Figure 28 – Transmission through DNH aperture over time as measured by the APD at 20 kHz acquisition rate. The untrapped, trapped, and trapped with applied magnetic field situations are shown. The solid blue line in the transmission time series is a moving average with a 2000-point window. Inset shows a typical autocorrelation curve of the transmission for a particle in the trapped state. ... 68 Figure 29 – a) to e) voltage distribution as measured by APD for a single trapping event under various applied fields. f) normalized fitting curves for 𝐻 = 0 kA/m (dotted) and 𝐻 = 24.16 kA/m (solid line). Notice the skewness towards higher transmission values with increasing applied magnetic field. ... 70 Figure 30 – Magnetic force acting on the trapped magnetite nanoparticle as a function of the applied magnetic field. ... 72 Figure 31 – a) calculated size distribution for magnetite nanoparticles based on 12 trapping events. b) measured size distribution of magnetite nanoparticles using scanning electron microscope. ... 74 Figure 32 – Wavefunctions of quantum harmonic oscillator corresponding to the lowest 6 energy states. The dashed black line indicates the potential well. ... 92 Figure 33 – Probability distributions of the vibrational states in a quadratic potential well as defined by the wavefunction... 93 Figure 34 – Typical double nanohole fabrication parameters. The black regions indicate the area that is exposed to the focused ion beam. The dwell time (dt) and number of passes (N) are 10 μs and 35 respectively. A 1-pixel × 50 nm line is drawn between the adjacent spheres to connect them. ... 94 Figure 35 – Double nanohole aperture fabricated with focused ion beam at 20 000 times magnification. Top: SEM image of DNH, imaging plane is perpendicular to surface normal. Bottom: tilted image of same aperture, imaging plant is tilted 30 ∘ to surface normal. ... 95 Figure 36 – Double nanohole aperture fabricated with focused ion beam at 40 000 times magnification. Top: SEM image of DNH, imaging plane is perpendicular to surface

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normal. Bottom: tilted image of same aperture, imaging plant is tilted 30 ∘ to surface normal. ... 96 Figure 37 – Double nanohole aperture fabricated with focused ion beam at 60 000 times magnification. Top: SEM image of DNH, imaging plane is perpendicular to surface normal. Bottom: tilted image of same aperture, imaging plant is tilted 30 ∘ to surface normal. ... 97 Figure 38 – Double nanohole aperture fabricated with focused ion beam at 80 000 times magnification. Top: SEM image of DNH, imaging plane is perpendicular to surface normal. Bottom: tilted image of same aperture, imaging plant is tilted 30 ∘ to surface normal. ... 98 Figure 39 – Double nanohole aperture fabricated with focused ion beam at 100 000 times magnification. Top: SEM image of DNH, imaging plane is perpendicular to surface normal. Bottom: tilted image of same aperture, imaging plant is tilted 30 ∘ to surface normal. ... 99 Figure 40 – Design parameters for C-shaped trapping aperture. ... 101 Figure 41 – Scanning electron microscope images of C-shaped trapping apertures. The size parameters for each aperture are given below the figures and correspond to the parameters set in Figure 40. The FIB parameters are the same as those described in “Appendix B – Fabrication of DNH at Different FIB Magnifications” with a beam magnification of 80 000 times. ... 101 Figure 42 – Normalized transmittance for parallel (left) and perpendicular (right) polarization, through a set of 21 C-shaped trapping apertures. Parameters A = 30 nm. Laser wavelength = 853 nm. ... 102 Figure 43 – Left: diagram illustrating the magnetization of a material (B) due to an applied magnetic field (H) for ferromagnetic, paramagnetic, and superparamagnetic materials. Right: material property domains for ferromagnetic particles as the size dimension approaches 0... 103 Figure 44 – Double potential well of a superparamagnetic state149. ... 104 Figure 45 – Vibrating sample magnetometer design. Sample is held in place within external magnetic field. Sample moves (vibrates) in sinusoidal fashion. The signal of vibrating sample is detected by coils and results in an AC current with amplitude proportional to the magnetic moment of the sample. ... 105

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Acknowledgments

Thank you to everyone in the lab, it has really been a pleasure to get to know you all. I want to thank Dr. Steeves for being on my committee, as well as for his support since my undergrad. Thank you to all of the UVic physics department; from lab instructors to professors, I have always valued their devotion to the students.

A special thanks to Dr. Reuven Gordon. I sincerely appreciate the opportunities you provided me with; and have thoroughly enjoyed my time working for you.

Cheers, Steve Thanks to: Daniel Darren Emilio Haitian Lewis Ryan Travis

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Dedication

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Chapter 1 – Introduction

Throughout history, humans have always used electromagnetic (EM) radiation to perceive the world. Sight, arguably the sense humans rely on most, is an evolutionary adaptation that allows the eye to detect a specific range of frequencies in the electromagnetic spectrum – i.e. visible light. An electromagnetic wave is an oscillation in electric and magnetic fields; this oscillation propagates forward at the speed of light (≈ 3 × 108_{m/s in vacuum). The}

properties of EM waves are determined by their frequency (𝑓) or wavelength (𝜆) which are intimately tied together via the following equation:

𝑐 = 𝜆𝑓 (1.1)

Depending on the frequency/wavelength, the interactions between light and matter can vary drastically. For example, visible light consists of wavelengths on the order of 𝜆 ≈ 400 × 10−9 m (violet) to 𝜆 ≈ 700 × 10−9 m (red); while the EM waves used to send signals to your car radio have wavelengths on the order of 𝜆 ≈ 100 m (AM radio) to 𝜆 ≈ 1 m (FM radio).

For scientists, light is a particularly useful tool for probing the characteristics of a substance. Optical microscopes for instance, allow scientists to image objects that are far too small to be seen with the naked eye. In a complementary class of optical methods, known as spectroscopy, scientists can observe the response of an object to different frequencies of incident light, and ascertain the molecular structure, temperature, and other properties from this information. As technology progresses, scientists continue to probe ever smaller molecules of interest. At present, optical methods are adept at characterizing

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molecules as small as several microns (~ 10−6 m); and cutting edge research is being conducted to push these capabilities into the nano regime (~ 10−9_m).

The remainder of this Chapter will provide some insight into current optical techniques that are employed to characterize or observe small molecules of interest. This Chapter will also present the current limitations of the methods and where there is room for improvement. The overall goal being considered in this thesis is efficient nanoparticle detection / sensing platforms, with the ultimate intention being to present a platform that is capable of single particle sensitivity and selectivity.

1.1 Optical Sensing

This Section introduces some notable label free optical techniques that are commonly used for sensing nanoscale particles. These techniques are, in general, complementary with respect to the properties of the analyte that the methods are able to resolve.

1.1.1 Dynamic Light Scattering

Dynamic light scattering is a method that uses fluctuations in the scattered intensity of coherent monochromatic light by an ensemble of small particles to determine the diffusion coefficient, and size distribution of the particles1–4. For particles much smaller than the wavelength of the incident light, the radiation scattered by the particle is effectively isotropic3. As a result, the intensity as measured at a specific scattering angle is dependent on the position of all of the particles involved. Over time the measured intensity of light will fluctuate due to changes in the position of the particles which is caused by Brownian motion. By taking the autocorrelation of the scattered intensity, the rate of Brownian motion can be determined. For a dilute solution of disperse nanoparticles the time constant

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of the exponential decay of the autocorrelation signal is inversely proportional to the diffusion coefficient, which is itself inversely proportional to the radius of the particle3–6_.

Dynamic light scattering is a useful technique to characterize the physical properties of a solution of nanoparticles in an in situ environment. By modifying the experiment, this technique can easily be extended to measured temperature effects on the diffusion coefficient, as well as to determine the extent of particle aggregation over time. While useful, this method requires a large amount of a priori knowledge of the particles being measured and their surrounding environment. Additionally, because the method relies on the intrinsic Brownian motion of the particles, dilute concentrations must be used so that interparticle collisions do not dominate particle motion. This method is not applicable to single particles.

1.1.2 Surface Plasmon Resonance Sensors

When monochromatic light in the visible frequency range interacts with the interface between a noble metal and a dielectric, surface waves of charge density oscillations are able to propagate7,8. These surface waves are known as surface plasmon polaritons. The coupling condition required for the formation of a surface wave is extremely sensitive to the electrical permittivity of the surrounding dielectric (typically a liquid)7,9. Changes in the coupling condition can be detected as a shift in the absorption spectra of the incident light – typically in terms of angle of incidence for monochromatic light, or in terms of wavelength for a fixed angle of incidence10–14. The most common form of this type of sensor is called the Kretschmann configuration, show in Figure 1.

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Figure 1 – Kretschmann configuration of a surface plasmon resonance sensor. Left: schematic

diagram of the measurement apparatus. Monochromatic light is focused onto a thin gold film via a prism. Surface plasmons are excited for a specific angle of incidence corresponding to an absorption peak as measured by the detector. As the analyte bonds to the functionalized surface of the gold, the effective index of the dielectric changes causing a shift in the plasmon resonance. Right: the angle of incidence corresponding to surface plasmon excitation as a function of time as the analyte if injected into the channel, saturates the sensing region, and is subsequently removed.

In the Kretschmann configuration a monochromatic light source is used to illuminate a thin noble metal film (typically gold) at a variety of angles of incidence. The coupling condition in this neutral configuration results in a narrow range of angles of incidence being absorbed by the metal in the form of a surface plasmon. In the typical implementation, the surface of the gold on the side of the microfluidic flow channel is functionalized to promote the desired analyte to bind to the surface. As the concentration of the analyte in the flow channel is increased, the binding rate to the surface also increases resulting in a change in the effective permittivity of the dielectric. This change in permittivity alters the coupling condition and results in a shift of the absorption spectrum (see Figure 1 right). As more analyte is flowed into the channel, the interaction between analyte and functionalized gold surface will reach equilibrium where binding and unbinding events occur at the same rate. Following this period, the analyte will be flushed out of the channel and the system is reset.

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Because of the ability of this method to easily probe the binding affinity between the analyte and the functionalization of the surface, it has found a plethora of applications within biology15–18. Surface plasmon resonance based sensors are a simple and relatively cheap testing apparatus. The method is often employed on ensembles; however recent advances in the field have shown the potential for single particle sensitivity19. Despite the high level of sensitivity and simple implementation, this method suffers from a lack of selectivity in terms of its ability to isolate single molecules for individual analysis.

1.1.3 Spectroscopy

Spectroscopy is an extremely useful and broad sensing technique that can provide complex information about the energy states of a molecule. Light interacting with a molecule will cause transitions between energy states which affects the transmitted or scattered signal. The energy of a molecule is defined by the electric, vibrational, and rotational energy states in descending order of energy magnitude. Probing the electronic energy states can provide information about the bonding between atoms in a molecule; while probing the vibrational states details the molecular structure and can act as a unique spectral fingerprint. For the purpose of this thesis the scope will be restricted to spectroscopic techniques that probe the vibrational modes of a molecule.

The energy of vibrational transitions in molecules typically corresponds to optical frequencies in the infrared region20. The two main spectroscopic methods used to probe the vibrational modes of molecules are infrared (IR) spectroscopy, and Raman spectroscopy. In IR spectroscopy a broad spectrum of IR light is radiated towards the analyte. The

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frequencies of light which correspond to vibrational transitions will be absorbed by the molecules and appear as absorption peaks in the transmission spectra. In Raman spectroscopy a monochromatic source is used to excite the analyte. Some of the scattering events will cause vibrational transitions and therefore the scattered light will gain or loose energy accordingly. These scatting events corresponds to spectral peaks that are shifted relative to the excitation frequency. While both IR and Raman spectroscopy deal with vibrational energy states they are in fact complimentary as certain vibrational modes are IR active while others are Raman active (and some are active in both)20,21. A more detailed investigation of the phenomena involved in Raman scattering is presented in Section “2.5 Raman Spectroscopy”.

Spectroscopy is an incredibly powerful technique in terms of the amount of information it can provide; however, it is also often a relatively weak process. Particularly in the case of nanoparticles, where the analyte is significantly smaller than the wavelength of incident light and the interaction cross section becomes quite small. As a result, spectroscopic techniques have typically been employed on ensembles. Modern advances to these techniques will be discussed in the following Section which show the ability to reach single particle sensitivity.

1.2 Nanostructured Enhancements

Presented here are current methodologies being employed to improve the abilities of some of the aforementioned optical sensing techniques. These methods primarily utilize nanostructures in noble metals to cause extreme electric field localization. This Section serves to present the current state of a select group of optical techniques which are

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progressing towards the ultimate limit of single particle sensitivity and selectivity. For a more thorough description of the interaction between electromagnetic waves and metal/dielectric interfaces see Section “2.3 Plasmons”.

1.2.1 Localised Surface Plasmon Resonance Sensors

Similar to the surface waves that were described in Section “1.1.2 Surface Plasmon Resonance”; metallic nanoparticles can also exhibit plasmonic resonances. These plasmonic resonances create intense electric field enhancements and simultaneously localize the electric field to small, subwavelength regions. A more thorough description of this phenomenon is presented in Section “2.3.2 Localized Surface Plasmons Resonances”. Because of these resonances, the metallic nanoparticles are extremely susceptible to changes in their surrounding and can be implemented as an effective small area sensing platform. The basic design of such a sensing apparatus is presented in Figure 2.

Figure 2 – Diagram illustrating the preparation and operation of a localized surface plasmon

resonance (LSPR) based biosensor based on changes in the surrounding dielectric permittivity. a) and b) metal nanoparticles are deposited onto a substrate. c) nanoparticles are functionalized to bind with desired analyte. d) analyte binds to nanoparticles causing a frequency shift in the plasmonic resonance – shown in e). This figure has been reprinted with permission from22_.

Recent advances in nanofabrication technologies have enabled advanced shapes such as stars or pyramids to be utilized for plasmonic resonance based sensing applications23–26. These complex nanoparticle shapes permit even larger electric field enhancements and

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localization. This technique is of current interest for biological applications due to its high sensitivity and low sample volume required22,27_{. These devices have been shown to detect}

analytes with extremely low concentrations in the nanomolar or subnanomolar ranges28,29. The small concentration and potential for single particle sensitivity make LSPR based sensing devices an intriguing technology for investigating small molecule interactions. Recent demonstrations of single biological particle detection with surface plasmon resonance based sensors include the detection of viruses and virus like particles30–33_{, and}

the detection of DNA functionalized gold nanoparticles19,34.

1.2.2 Surface Enhanced Raman Spectroscopy

Similar to the other methods presented in this Section, the field localization caused by plasmonic effects can also increase the intensity of the intrinsically weak Raman scattering signal20,35,36_{. Because of the small particle sizes with respect to the excitation light, the}

Raman scattering cross section is particularly small for low concentration nanoparticle analytes. By using nanostructured metals, plasmonic effects can increase the interaction between the electric field and the analyte. This effect is enhanced further if the particle can be localized to the immediate vicinity of the electric field enhancement. A more detailed explanation of the phenomenon is presented in Section “2.5.1 Surface Enhanced Raman Spectroscopy”. Recent works have demonstrated nanoplasmonic platforms that increase the Raman interaction, resulting in a single particle sensitivity37.

1.2.3 Extraordinary Optical Transmission

For a periodic array of nanoapertures in a thin metal film, at certain frequencies the transmission through the array is much greater than predicted by classical

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electromagnetism38,39. This phenomena, known as extraordinary optical transmission (EOT), has been directly observed to occur as the result of plasmonic effects40_{. The spectral}

location of the transmission resonances is defined by the periodicity of the arrays41. The ability to tune the transmission properties of the array makes EOT based sensing platforms an intriguing option for nanoscale optical sensing. Past works have shown the ability for these structures to be employed in refractive index sensing (i.e. SPR)42,43, as well as in spectroscopic applications44–46_.

1.3 Optical Trapping

In order to isolate a single particle (single particle selectivity), a method has been developed that uses purely optical forces to manipulate small particles, i.e. optical trapping47_{. The}

physical phenomena that govern this process are detailed in Section “2.2 Rayleigh Scattering” and Section “2.4 Optical Trapping”. Due to the minute nature of optical forces, this method has historically been relegated to the micro regime. This is due to the fact that micron sized particles are small enough to be easily influenced, yet large enough to interact strongly with light. It is currently of great interest to advance the method of optical trapping towards the nanoscale. For the purposes of this work, focus is placed on the forefront of this research field as it applies to objects with nanoscale dimensions.

The method of optical trapping has been used to investigate the Brownian motion of trapped nanoparticles48_{, plasmonic coupling between gold nanospheres}49_{, and nonlinear}

trapping effects from optical forces50. Optical traps have also been demonstrated to directly manipulate nanostructures; one example is an optically driven rotary motor that is capable of spinning at frequencies up to 42 kHz51. Trapped particles can also be optically excited

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such as the generation of unconverted photons from KNbO3 nanowires52. By using multiple

trapping frequencies, optical trapping setups have been used to passively sort gold nanoparticles based on their size53. In biological applications, low power optical traps have been utilized to observe single molecule protein binding kinetics, and probe low frequency vibrational modes of proteins54,55. Perturbations of the method have also been developed, such as holographic optical traps and optoelectronic traps, which demonstrate the ability for high throughput nanoassembly and fabrication56,57_.

1.4 Contributions of this Thesis

In this thesis, work is presented which simultaneously demonstrates single particle sensitivity and isolation, allowing for data to be obtained from an individual analyte nanoparticle. Building on prior experiments, which have used the method of aperture assisted optical trapping to manipulate and detect small objects, presented here are two new methods that are able to independently determine the characteristic properties of an isolated nanoparticle. In the first work, an apparatus is developed which looks at the reflection of light by a nanoparticle to obtain information about the vibrational modes of its constituent molecules. In this way, a “spectral fingerprint” of the molecule is obtained, this enables the unique identification of the molecular structure of the particle. In the second method, a superparamagnetic nanoparticle is subjected to varying applied magnetic fields while in an optical trap. By observing the response of the particle to the applied magnetic field, the complex refractive index, magnetic susceptibility, remanence, and size distribution can be determined. The methods presented here allow researchers new avenues to identify and characterize individual nanoparticles in a label free method with a minimum of a priori knowledge of the properties of the analyte.

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The following Chapter will build the necessary theoretical foundation required to understand the experimental results presented in this work. Following the theory, an outline of the techniques used in aperture assisted trapping will be detailed. In Chapter 4 and 5, two methods for label free characterization of individual nanoparticles will be presented. Chapter 4 deals specifically with an experimental demonstration of plasmon enhanced Raman spectroscopy on an individually trapped nanoparticle; the spectra for two different material nanoparticles is presented. In Chapter 5, the method of optical trapping magnetic field (OTMF) is reported. In OTMF an external magnetic field is applied to a trapped magnetic nanoparticle, by varying the magnetic field and observing the influence on the trapped particle, characteristic properties of the magnetic nanoparticle are ascertained. Following the conclusion (Chapter 6), a brief forecast on the future of aperture assisted trapping is discussed is Chapter 7.

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Chapter 2 – Theory

2.1 Introduction

In this Chapter the underlying principles of the techniques used in this work will be described. The ultimate goal is to outline how optical methods can be used to isolate and characterize nanoparticles. Optical trapping is a method that uses the intensity profile of light to manipulate small objects. This phenomenon is based on the conservation of momentum that occurs as light is scattered by a particle; and by the relationship between the optical response of the particle and that of the surrounding medium (characterized by the dielectric permittivity). The interaction between light and subwavelength particles is described in Section “2.2 Rayleigh Scattering”.

In Section “2.3 Plasmons,” the interaction between light and noble metals at optical frequencies will be detailed. The interesting features of this interaction results from charge density oscillations on a metal’s surface and are influenced by the surrounding dielectric. Extreme field localization is associated with plasmon oscillations which can be further enhanced by small metallic particles and structures. The effects of these plasmon resonances are integral to the aperture assisted trapping works presented here.

Section “2.4 Optical Trapping” presents the basic principles of optical trapping and how this phenomenon can be understood. In order to trap dielectric nanoparticles, complimentary methods must be employed. Such methods include aperture assisted trapping which can be visualized by analysing Bethe’s theory of transmission through small holes.

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Finally, in Section “2.5 Raman Spectroscopy,” a method of using the scattering of light by a molecule as a spectral fingerprint is analyzed. This form of spectroscopy is based on transitions between the vibrational modes of a molecule. Raman spectroscopy allows scientists to noninvasively identify the molecular structure of a particle based on the scattered light. This spectral fingerprint is an extraordinarily useful technique and its efficiency can be greatly enhanced by using plasmons to localize the electric field to subwavelength dimensions.

2.2 Rayleigh Scattering

Rayleigh scattering is a domain of electromagnetic theory that is concerned with the interaction of light with subwavelength particles, specifically for elastic scattering events (as opposed to inelastic scattering such a Raman scattering as described in “2.5 Raman Spectroscopy”). In the context of optical trapping there are two main components of optical forces that must be considered: scattering force, and gradient force. Using Rayleigh scattering theory this thesis will outline how optical forces act on small dielectric spheres as an illustrative example for how light can be used to manipulate subwavelength particles.

2.2.1 Scattering Force

Consider a small sphere which is illuminated by a collimated beam of monochromatic, linearly polarized light. In the case where the sphere is small enough, at any instant in time the electric field surrounding the sphere is approximately constant across its volume. This approximation implies that the interaction is effectively electrostatic in nature. The sphere becomes polarized as the electrons are displaced from their equilibrium position around their respective nuclei. In this case the interior of the sphere experiences a uniform electric field (see Figure 3) according to:58

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𝐸internal= (

3𝜖particle

𝜖particle+ 2𝜖med) 𝐸0 (2.2.1)

Here, 𝐸₀ is the amplitude of the incident electric field and 𝐸internal is the amplitude of the

electric field within the sphere. 𝜖_med and 𝜖_particle are the dielectric permittivities of the surrounding medium and sphere respectively.

Figure 3 – Dipole moment induced in a subwavelength dielectric sphere by an applied electric

field. Notice the electric field lines are parallel within the sphere, i.e. constant E-field.

The displacement of the electrons within the sphere from their equilibrium positions’ results in an induced dipole moment of the sphere. The external electric field is a superposition of the field in the absence of the sphere, and the field radiating from an electric dipole placed at the centre of the sphere. The dipole moment induced in the sphere is given by59

𝑝 = 𝛼𝐸0 where 𝛼 = 4𝜋𝜖med𝑟3(

𝜖_particle− 𝜖_med

𝜖particle+ 2𝜖med) (2.2.2)

Here, 𝑝 is the dipole moment induced in a dielectric sphere of radius 𝑟, and 𝛼 is the polarizability of the sphere. For an applied electric field oscillating in a sinusoidal fashion,

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the dipole moment of the spherical dielectric particle acts as an electric dipole oscillating at the frequency of the incident light. The scattering intensity is given by:58

𝐼 =16𝜋 4_𝑟6 𝑅2_𝜆4 ( 𝜖particle− 𝜖med 𝜖_particle+ 2𝜖_med) 2 sin2_(𝜃)𝐼 0 (2.2.3)

Where 𝑅 is the distance from the centre of the sphere (dipole) to the point where the intensity is measured, and 𝜃 is the angle between the vector 𝑅⃑ and the axis of electric polarization (see Figure 4).

Figure 4 – Radiation pattern of dielectric sphere (dipole). The radial distance from any point on

the surface of the wavefront (red surface) to the origin, is proportional to the intensity scattered in that direction.

By integrating equation 2.2.3 over the surface of a sphere of radius 𝑅 > 𝑟 we can obtain the total power radiated by the sphere, i.e.

𝑷scattered = ∫ 𝐼 ⋅ 𝑑𝑆𝑟= ∫ ∫ 16𝜋4𝑟6 𝑅2_𝜆4 ( 𝜖particle− 𝜖med 𝜖particle+ 2𝜖med) 2 sin2(𝜃)𝐼0⋅ 𝑅2sin(𝜃)𝑑𝜃𝑑𝜙 𝜋 0 2𝜋 0

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𝑷scattered = 128𝜋5𝑟6 3𝜆4 ( 𝜖particle− 𝜖med 𝜖particle+ 2𝜖med) 2 𝐼0 (2.2.4)

Under this electric dipole approximation, the scattered power is radiating isotropically. As a result, the total flux pattern of electromagnetic radiation has changed. Conservation of momentum thus requires that the particle move in parallel with the initial path of the light. The force acting on the particle due to scattering is59–61

𝐹scattered = √𝜖med/𝜖0 𝑐 ⋅ 𝑷scattered (2.2.5) 𝐹scattered = 128𝜋5_𝑟6 3𝜆4 √𝜖_𝜖med 0 𝑐 ( 𝜖particle− 𝜖med 𝜖particle+ 2𝜖med) 2 𝐼₀ (2.2.6)

Or as it is more colloquially written, in terms of the refractive index:

𝐹scattered = 128𝜋5𝑟6 3𝜆4 𝑛med 𝑐 ( 𝑚2− 1 𝑚2_{+ 2}) 2 𝐼0 where 𝑚 = 𝑛particle 𝑛med = √ 𝜖particle 𝜖med (2.2.7)

The important factor to observe here is that the scattering pressure (force per unit area) is proportional to the incident intensity, and to (𝑟/𝜆)4. This implies that as the particles radius is scaled below the wavelength of incident light, the magnitude of the scattering force will diminish rapidly.

2.2.2 Gradient Force

Continuing with the Rayleigh description of light interacting with subwavelength particles, we can investigate how the induced dipole moment of the particle interacts with the intensity distribution of light, known as the gradient force. The force that the particle experiences is ultimately the Lorentz force, i.e.62,63

𝐹 = (𝑝 ⋅ ∇)𝐸 +𝑑𝑝

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𝐹 = 𝛼 ((𝐸 ⋅ ∇)𝐸 +𝜕𝐸

𝜕𝑡 × 𝐵) (2.2.9)

To realize this relationship in a more descriptive form we use the following vector identity in conjunction with Faraday’s law from Maxwell’s equations63

∇(𝐴 ⋅ 𝐵) = 𝐴 × (∇ × 𝐵) + 𝐵 × (∇ × 𝐴) + (𝐴 ⋅ ∇)𝐵 + (𝐵 ⋅ ∇)𝐴 (2.2.10) with 𝐴 = 𝐵 = 𝐸 this becomes:

1 2∇𝐸

2 _{= 𝐸 × (∇ × 𝐸) + (E ⋅ ∇)E} _(2.2.11)

Faraday's law ∇ × 𝐸 = −𝜕𝐵

𝜕𝑡 (2.2.12)

Using Faraday’s law and the above vector identity, the force on the particle can be simplified to

𝐹 =1 2𝛼∇𝐸

2_{+ 𝛼} 𝜕

𝜕𝑡(𝐸 × 𝐵) (2.2.13)

By taking the time average of the above expression we get 𝐹grad = 〈𝐹〉 = 1 4𝛼∇|𝐸| 2_{+ 𝛼𝜇} 0 𝜕𝐼₀ 𝜕𝑡 (2.2.14)

For the case of time-constant optical intensities, this reduces to61

𝐹_grad =1 4𝛼∇𝐸 2_{with 𝛼 = 4𝜋𝜖} med𝑟3( 𝜖particle− 𝜖med 𝜖_particle+ 2𝜖_med) (2.2.15)

This formula describes the way in which a subwavelength particle responds to spatially varying electric field intensity distributions. The term in brackets in the polarizability expression (equation 2.2.14) is known as the Clausius-Mossotti factor and illustrates how the polarizability of a particle is influenced by the surrounding medium. For the case of a dielectric particle with higher relative permittivity, the gradient force acts to pull the particle to regions of higher electric field intensities. This situation can be thought of as a

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particle of higher optical density being swept towards higher electric fields, while less optically dense materials are expelled from those regions. From the analytical form of the gradient force, we can see that it is proportional to the third power of the particles radius. When compared to the scattering force, which is proportional to the sixth power of the particles radius (and inversely proportional to the fourth power of the wavelength) we can see that for deeply subwavelength particles, the gradient force becomes dominant.

2.3 Plasmons

The field of plasmonics is relatively new, having only been developed into its current form around the 1970’s64_{. The effects of plasmonics however, have been observed for centuries,}

with artists incorporating trace amounts of metallic nanoparticles into glass to create unique and visually stunning effects65 (see Figure 5). In its essence, plasmonics is a theory which describes collective electron density oscillations in a noble metal under the influence of electromagnetic radiation. The first experimental observation of plasmonic effects took place in 1902 when examining the diffraction of light by metal grating, a phenomena which could not be explained for more than half a century65,66_{. Since its initial development the}

field of plasmonics has matured into an extensive engineering and research area which describes the unique electric field distributions that occur at metal-dielectric interfaces around the visible frequency range67–70.

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Figure 5 – Lycurgus cup, a Roman artifact dating back to the 4th_{century AD. The glass contains} trace amount of silver and gold nanoparticles which exhibit plasmon resonances, thereby affecting the transmission and reflection of light. The image on the left shows the cup transmitting light, while the figure on the right is viewed mainly through reflection. Reprinted with permission from71_.

To understand plasmonic interactions we must first describe the way in which a noble metal optically responds to light as a function of frequency. Specifically, we must define the permittivity 𝜖(𝜔) as a function of frequency. In order to describe the permittivity, we start our analysis by considering the free electron gas model of noble metals. In this approximation the electron motion is defined by the following second degree differential equation7 𝑚𝑑 2_𝑥 𝑑𝑡2 + 𝑚𝛾 𝑑𝑥 𝑑𝑡 = −𝑒𝐸 (2.3.1)

Where 𝑚 is the mass of the electrons, 𝑥 is their position, 𝛾 is a damping factor resulting from collisions, 𝑒 is the charge of an electron, and 𝐸 is the applied electric field. By

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assuming the form for the applied electric field, i.e. 𝐸(𝑡) = 𝐸0e−𝑖𝜔𝑡, the motion of

electrons can be written as

𝑥(𝑡) = 𝑒

𝑚(𝜔2_{+ 𝑖𝛾𝜔)}𝐸0e−𝑖𝜔𝑡 (2.3.2)

The polarization density (𝑃 = 𝑞𝑑) can then be described by 𝑃 = −𝑁𝑒𝑥 = − 𝑁𝑒

2

𝑚(𝜔2_{+ 𝑖𝛾𝜔)}𝐸 (2.3.3)

Here, 𝑁 is the density of electrons. The displacement field in this case becomes

𝐷 = 𝜖₀𝐸 + 𝑃 = 𝜖₀(1 − 𝜔𝑝 2 𝜔2_{+ 𝑖𝛾𝜔}) 𝐸 with 𝜔𝑝= √ 𝑁𝑒2 𝜖0𝑚 (2.3.4)

The constant 𝜔_𝑝 is known as the plasma frequency. Using an alternate definition for the displacement field, 𝐷 = 𝜖₀𝜖_𝑟(𝜔)𝐸, we define the dielectric function of this free electron gas

𝜖_𝑟(𝜔) = 1 − 𝜔𝑝

2

𝜔2_{+ 𝑖𝛾𝜔} (2.3.5)

For real metals, the above relation must be adjusted to account for the response to frequencies much greater than the plasma frequency7_{, i.e.}

𝜖_𝑟(𝜔) = 𝜖_∞− 𝜔𝑝

2

𝜔2_{+ 𝑖𝛾𝜔} (2.3.6)

Here we have arrived at the Drude model for the dielectric response of a metal which is our starting point for the investigation of plasmonic effects.

The simplest type of plasmonic resonance is called a volume plasmon and is possible for the case of a free electron gas with negligible damping. At the plasma frequency the dielectric response of an ideal metal is exactly zero (i.e. 𝜖_𝑟(𝜔_𝑝) = 0 for 𝛾 = 0). Therefore,

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the displacement field is also zero causing the polarization and electric field to be related by:

𝑃 = −𝜖₀𝐸 or 𝐸 = − 𝑃

𝜖₀ (2.3.7)

In other words, the field induced within the material is a pure depolarization field and the collective charge density oscillations of the electron gas are exactly out of phase with the incident electric field.

2.3.1 Surface Plasmon Polaritons

A more interesting and useful example of plasmon oscillations occurs at the interface between a metal and a dielectric. In this situation photons can be coupled to plasma oscillations in the metal allowing a type of surface wave to propagate7. To start our analysis we first present Maxwell’s equations in their differential form, i.e.63

Gauss' law ∇ ⋅ 𝐷 = 𝜌_𝑒𝑥𝑡 (2.3.8a)

no name ∇ ⋅ 𝐵 = 0 (2.3.8b)

Faraday's law ∇ × 𝐸 = −𝜕𝐵

𝜕𝑡 (2.3.8c)

Ampere's law with Maxwell's correction ∇ × 𝐻 = 𝐽𝑒𝑥𝑡+

𝜕𝐷

𝜕𝑡 (2.3.8d)

Where 𝐷 = 𝜖0𝐸 + 𝑃. By assuming no external charges or currents (𝜌 = 0, 𝐽 = 0) and

combining the curl equations with the vector identity ∇ × ∇ × 𝐸 = ∇(∇ ⋅ 𝐸) − ∇2_𝐸

(equation 2.2.10) we obtain: ∇ (−1 𝜖𝐸 ⋅ ∇𝜖) − ∇ 2_{𝐸 = −𝜇} 0𝜖0𝜖𝑟 𝜕2_𝐸 𝜕𝑡2 (2.3.9)

If the permittivity is translationally invariant (i.e. ∇𝜖 = 0) this results in the standard wave equation for electromagnetism7

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∇2𝐸 = 𝜖 𝑐2

𝜕2𝐸

𝜕𝑡2 (2.3.10)

By assuming the form of the applied electric field as a plane wave (i.e. 𝐸 ~ e−𝑖𝜔𝑡) and making use of the definition for the wave vector (𝑘₀ = 2𝜋/𝜆₀ = 𝜔/𝑐) then the wave equation simplifies to:

∇2_{𝐸 + 𝜖𝑘}

02𝐸 = 0 (2.3.11)

To investigate the electromagnetic interactions at a metal-dielectric interface we define a system with light propagating in the x-direction, and where a metal exists for all 𝑧 < 0 (𝜖_𝑟 defined by Drude model), and a dielectric for all 𝑧 > 0 (𝜖_𝑟 = constant). The z-component of the wave equation is

𝜕2_𝐸 𝜕𝑧2 + 𝑘𝑧 2_{𝐸 =}𝜕2𝐸 𝜕𝑧2 + (𝜖𝑘0 2_{− 𝑘} 𝑥2)𝐸 = 0 (2.3.12)

We can also now write Faraday’s law and Ampere’s law from Maxwell’s equations as ∇ × 𝐸 = 𝑖𝜔𝜇0𝐻 and ∇ × 𝐻 = −𝑖𝜔𝜖0𝜖𝑟𝐸 (2.3.13)

Note that by virtue of the problem definition, the system is translationally invariant along the y-axis, and therefore all derivatives with respect to 𝑦 are exactly zero. For this system of equations there are two forms of the solution, one corresponding to TM propagating waves where 𝐸_𝑥, 𝐸_𝑧, and 𝐻_𝑦 are all non-zero, and another for TE propagating waves with 𝐻_𝑥, 𝐻_𝑧, and 𝐸_𝑦 all non-zero. For the case of TE polarization it can be shown that no surface modes exist and thus for the study of surface plasmon polaritons we will restrict further analysis to the TM case7.

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𝜕2𝐻_𝑦 𝜕𝑧2 + (𝜖𝑘02− 𝑘𝑥2)𝐻𝑦 = 0 (2.3.14a) 𝐸_𝑥 = − 𝑖 𝜔𝜖₀𝜖_𝑟 𝜕𝐻_𝑦 𝜕𝑧 (2.3.14b) 𝐸𝑧 = − 𝑘_𝑥 𝜔𝜖0𝜖𝑟 𝐻𝑦 (2.3.14𝑐)

Solving the above equations and then applying boundary conditions at 𝑧 = 0 we obtain

𝐻𝑦 = 𝐶0e𝑖(𝑘𝑥𝑥−𝑘𝑧𝑧) (2.3.15a) 𝐸𝑥 = 𝐶0 𝑖𝑘_𝑧 𝜔𝜖₀𝜖_𝑟e 𝑖(𝑘𝑥𝑥−𝑘𝑧𝑧) (2.3.15b) 𝐸_𝑧 = −𝐶₀ 𝑘𝑥 𝜔𝜖0𝜖𝑟 e𝑖(𝑘𝑥𝑥−𝑘𝑧𝑧) _(2.3.15c)

It is important to note that both 𝜖_𝑟 and 𝑘_𝑧 are functions of 𝑧 and have a discrete discontinuity at the boundary 𝑧 = 0. Explicitly, for the problem defined here this implies

𝜖_𝑟(𝑧 < 0) = 𝜖(𝜔) (Drude model) 𝜖_𝑟(𝑧 > 0) = constant 𝑘_𝑧,𝑧<0 = √𝜖_𝑟,𝑧<0𝑘₀2− 𝑘_𝑥2 _(2.3.16a) 𝑘_𝑧,𝑧>0 = √𝜖_𝑟,𝑧>0𝑘₀2_{− 𝑘} 𝑥 2 _(2.3.16b) 𝑘𝑧,𝑧<0 𝑘_𝑧,𝑧>0 = − 𝜖𝑟,𝑧<0 𝜖_𝑟,𝑧>0 (2.3.16c)

Using these results we are able to obtain the dispersion relation for surface plasmons at this metal dielectric interface (see Figure 6).

𝑘𝑥= 𝑘0√

𝜖_𝑎𝜖_𝑏 𝜖𝑎+ 𝜖𝑏

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Where 𝜖𝑎 and 𝜖𝑏 are the relative permittivities of the materials for 𝑧 < 0 and 𝑧 > 0

respectively (note that 𝜖_𝑎 = 𝜖_𝑟,𝑧<0(𝜔) is frequency dependent).

Figure 6 – Dispersion relation for metal-dielectric interface using the Drude model7_(𝜖

∞ = 1,

𝜖𝑟,𝑧>0= 1). Left figure indicates the ideal lossless case, while the figure on the right includes losses

(𝛾 = 0.1). The solid blue lines correspond to the real part of the propagation wave vector while the dashed blue lines are the imaginary part. The red line indicates the lossless surface plasmon polaritons frequency, and the solid black line is the light line showing the dispersion relation for a plane wave propagating in a vacuum (𝜖𝑟 = 1).

Figure 6 shows the relation between the propagation wave vector (𝑘_𝑥) and the applied electromagnetic frequency relative to the plasma frequency. At the resonance frequency of equation 2.3.17, waves are able to propagate with a disproportionately large wave vector along the metal-dielectric interface. As this surface plasmon wave propagates, the electric field intensity is confined close to the surface of the metal. For frequencies significantly above the plasma frequency the metal acts as a dielectric.

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2.3.2 Localized Surface Plasmons Resonances

For the case of isolated or partially isolated subwavelength metallic particles, another interesting phenomenon is observed. To understand this, we look at the dipole moment induced in a small metallic sphere, i.e.

𝑝 = 𝛼𝐸₀ where 𝛼 = 4𝜋𝜖med𝑟3(

𝜖particle− 𝜖med

𝜖_particle+ 2𝜖_med) (2.3.18) Where 𝜖particle and 𝜖med are the permittivities of the metallic nanoparticle and surrounding

dielectric medium respectively.

For the case of metallic nanoparticles, we observe that their dielectric permittivity can take on negative values, and as 𝜖particle → −2𝜖med (known as the Fröhlich condition72) the

polarizability and therefore induced dipole moment approaches infinity in the lossless case. Damping of charge density oscillations in the metal prevents infinite values from being obtained in reality, but this serves to illustrate the extreme electric field enhancements that these particles are able to provide. As an extension of this, metallic nanoparticles that are nearby can strongly couple to one another, further extending the electric field enhancement capabilities (see Figure 7)73_.

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Figure 7 – Coupling between nearby nanoparticles and relative field enhancement with respect to

electric field orientation. a) shows two small metallic particles with their relative position vector perpendicular to the applied electric field and negligible coupling. b) indicates the case where the nanoparticles are orientated in parallel with the applied electric field and a large electric field enhancement is observed between the two particles, known as a plasmonic “hot spot”.

Figure 8 – Plasmonic resonances for gold nanoparticles as function of interparticle distance. Left:

average optical cross section for coupled gold nanoparticles (20 nm spheres – top/red, ellipsoids with aspect ration = 2 – bottom/blue). Right: total cross section normalized to physical cross section for gold nanospheres and ellipsoids in various dielectric mediums. Reprinted with permission from22_.

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In the subwavelength approximation, the metallic sphere is treated as a dipole with interaction cross sections given by:8

𝜎scattering= 3 2𝜋( 𝜔 𝑐) 4 (𝜖med 𝜖₀ ) 2 𝑉2 (𝜖particle ′ _{− 𝜖} med) 2 + (𝜖_particle′′ )2 (𝜖_particle′ + 2𝜖med) 2 + (𝜖_particle′′ )2 (2.3.19a) 𝜎extinction = 9𝜔 𝑐 ( 𝜖_med 𝜖₀ ) 3 2 𝑉 𝜖particle ′′ (𝜖_particle′ + 2𝜖med) 2 + (𝜖_particle′′ )2 (2.3.19b) 𝜎absorbed= 𝜎extinction− 𝜎scattering (2.3.19c)

Where 𝜎 denotes the respective cross sections for scattering, extinction, and absorption, 𝑉 is the volume of the particle, and 𝜖′, 𝜖′′ denote the real and imaginary components of the permittivity. As shown above, when the real part of the permittivity of the particle approaches negative two times that of the surrounding medium, the interaction with light increases drastically. This condition for this phenomenon is similar in form to the condition of surface plasmons presented earlier.

2.4 Optical Trapping 2.4.1 Introduction

Optical trapping is a method by which electromagnetic radiation can be used to impart a force on a particle in such a way that the particle is localized in two or three dimensions. The concept of optical trapping can be understood for larger particles using the ray optics model. Consider a situation where a photon of light scatters off of a particle. Because photons carry linear momentum, any change in the direction of photons will require a force to be imparted on the particle such that momentum is conserved, a phenomenon commonly known as radiation pressure. In a similar way, angular momentum can also be exchanged

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between a particle and photon51,74. The theoretical understanding of these optical forces was developed during the 19th_century75,76_.

The first demonstration of radiation pressure used thermal light sources to deflect a mirror attached to a torsion balance from its equilibrium position77–79. At the time it was believed that due to the minuteness of these forces they could not be utilized in any practical applications79,80. It was not until the advent of the laser that optical forces had the magnitude necessary to be utilized as an effective tool. In 1970 Arthur Ashkin et al. developed the first experimental realization of optical trapping47,60,81,82. Since then the method of optical trapping has found a plethora of applications in manipulation51,83–85, assembly53,56,86, and sensing49,54,87–89 of micro and nanoscale objects.

Shown in Figure 9a) is a plot of the typical forces a particle will experience due to optical interactions when located at or near the beam waist of a highly focused laser beam. The two main components of the total optical force are the scattering and gradient force. In the ray optics approximation (which applies to particles with size much greater than trapping wavelength) the scattering force is due to high angle scattering events (reflection), where photons will push the particle along the direction of the laser beam’s k-vector. Conversely the gradient force pulls or pushes the particle depending on if it focuses or defocuses the light. The gradient force can be visualized as a result of refraction using the ray optics approximation as shown in Figure 9b). The total force (dashed black line in Figure 9a)) is the sum of both the scattering and gradient forces, anywhere that the total force is zero and surrounded by restoring forces is theoretically a potential trapping location. Figure 9 c) shows a schematic diagram indicating several different particle locations and the forces acting on the particle.

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Figure 9 – a) graph showing the forces acting on a particle in an optical trapping configuration for

a Gaussian beam profile. The point x = 0 at the intersection of the axis, coincides with the centre of the focused beam waist (dashed line in c)). b) a simplified force vector diagram of the ray-model approximation to trapping forces, the length of each vector indicates the relative intensity of that interaction. c) a schematic diagram of an optical trapping configuration. The circles are particles near the beam waist showing relative intensities of scattering, gradient, and net forces for each particle. The particle just above the dashed line indicates the stable trapping position as indicated in a).

For Rayleigh particles, that are much smaller than the trapping wavelength, significantly greater optical powers are required for a stable free-space trap and the interactions are governed by the subwavelength quasi-static methods developed in Section “2.2 Rayleigh Scattering”. The reason for requiring higher optical intensities for smaller particles (even through scattering forces are negligible), is due to an increase in Brownian motion at

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smaller size scales90. For a spherical dielectric particle in an optical trap, the trapping potential is91 𝑈 =2𝜋𝑛med𝑟 3 𝑐 ( 𝑛particle− 𝑛med 𝑛particle+ 2𝑛med) 𝐼 (2.4.1)

Where 𝑛med and 𝑛particle are the refractive index of the surrounding medium and particle

respectively, 𝑟 is the particle radius, and 𝐼 is the intensity of light. In order for an optical trap to be stable, the potential well of the trapping force must significantly exceed the thermal kinetic energy of the particle so that the particle is not lost due to Brownian motion. As can be seen in the equation above, the trapping potential is proportional to the particles radius cubed. This implies that if the radius of the particle is reduced by one order of magnitude, then the intensity of the trapping laser must be increased by three orders of magnitude to maintain the same trapping potential. For these reasons it is exceedingly difficult to trap most dielectric particles on the nanometer scale without using excessive laser powers which would damage the particle, particularly in the case of biological samples.

To resolve this limitation, researchers have turned to new trapping techniques such as aperture assisted trapping. This method enables the effective trapping of dielectric and biological particles in the nanometre regime. The following sections serve to provide some insight into the phenomena that permits aperture assisted trapping to be an effective optical tool.

2.4.2 Bethe’s Aperture Theory

To understand how an aperture can be used to trap subwavelength particles, we must know how the apertures themselves interact with light. The theoretical understanding for

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subwavelength aperture transmission characteristics was developed by Hans Bethe in 194492_{. Bethe’s initial work used a quasi-static approximation to the transmission of a plane}

wave through a subwavelength circular aperture in an infinitely thin perfect electrical conductor film. In this approximation, the circular aperture is replaced with a magnetic dipole. The power radiated (𝑷) by the dipole is given by:93

𝑷 =𝑍0𝑘

4

12𝜋 |𝑝𝑚|

2_{with 𝑝}

𝑚= 𝛼𝑚𝐻0 (2.4.2)

Here, 𝑍0 is the impedance of free space, 𝑘 = 2𝜋/𝜆 is the wave vector, and 𝑝𝑚 is the

magnetic dipole moment of the aperture. For a circular aperture of radius 𝑟_𝑎 in a thin perfect electrical conductor, the net magnetic polarizability is94

𝛼_𝑚 =8𝑟𝑎

3

3 (2.4.3)

The transmission through the aperture is then half of the total power radiated by the dipole,

i.e. 𝑇 = 128𝑍0𝜋 3 27 𝑟_𝑎6 𝜆4𝐻0 (2.4.4)

As can be seen above, the transmission through the aperture is proportional to the sixth power of the radius of the aperture. When normalized to the area of the aperture the transmittance is proportional to (𝑟

𝜆) 4

, indicating a rapid decrease in transmittance as the aperture is scaled below the wavelength of incident light. In the case where the refractive index of the aperture and surrounding medium is replaced from material 1 to material 2, the relative change in transmission is

𝑇₂ 𝑇₁ = ( 𝜆₁ 𝜆₂) 4 = (𝑛2 𝑛₁) 4 (2.4.5)

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The above equation indicates how increasing the effective index of the medium can cause a large increase in transmission as illustrated in Figure 10.

Figure 10 – Illustrative example of transmission change through a subwavelength aperture upon

dielectric loading which increases the effective refractive index. a) and b) show the transmission through the aperture without and with an embedded particle respectively. c) indicates the transmission curves as a function of wavelength.

Work has been conducted to extend this theoretical approximation to real metallic films at visible frequencies where the perfect electrical conductor and infinitely thin approximations become invalid. Experimental evidence shows a strong correlation for these real apertures when compared to the theory developed by Bethe as shown in Figure 11 below95. This implies that although the axioms assumed in Bethe’s theory are